In past years we have derived an exact probability density for the three- phase invariant which is a widely used tool in direct methods of phase determination in crystallography. The numerical calculation of this function requires a nearly prohibitive use of computer time. Consequently numerical techniques have been developed which allow one to find accurate approximations to the exact results with minimal computing effort, and which are far more accurate than presently available approximations in crystallography. These have been applied to the P1 space group. Calculations are currently in progress to find numerical values of the corresponding probability density for the P2 space group. Initial measurements have been made of background noise in crystallographic measurements. These show that the usually made assumption of a Poisson distribution of the noise is only accurate at large scattering angles, but at lower ones a combination of Poisson distributions is required to characterize the data. Experiments have been initiated to see whether this has a significant effect on structure determination.